Radiative decays of the spin-$2$ partner of $X(3872)$

Abstract

It has been generally expected that the $X(3872)$ has a spin-2 partner, $X_2$, with quantum numbers $J^{PC}=2^{++}$. In the hadronic molecular model, its mass was predicted to be below the $D^{\ast }{\bar{D}}^{\ast }$ threshold, and the new structure reported in the $\gamma \psi (2S)$ invariant mass distribution by the Belle Collaboration with mass $M=(4014.3\pm 4.0\pm 1.5)$ MeV and decay width $\Gamma =(4\pm 11\pm 6)$ MeV, with a global significance of 2.8 $\sigma$, is a nice candidate for it. We consider the radiative decay widths for the $X_2\to \gamma \psi$ with $\psi =J/\psi ,\psi (2S)$ treating the $X_2$ as a $D^{\ast }{\bar{D}}^{\ast }$ shallow bound state, and estimate the events of $X_2$ in two-photon collisions that can be collected in the $\gamma J/\psi \to \gamma {\ell }^{+}{\ell }^{-}$ ($\ell =e,\mu$) final states at Belle. Based on the upper limit for the ratio of decay widths of $X(3872)\to \gamma \psi (2S)$ and $X(3872)\to \gamma J/\psi$ measured by BESIII, we predict the similar ratio $\Gamma (X_2\to \gamma \psi (2S))/\Gamma (X_2\to \gamma J/\psi )$ to be smaller than $1.0$. We suggest searching for the $X_2$ signal in the $\gamma J/\psi$ invariant mass distribution via two-photon fusions. The results will lead to insights into both the $X(3872)$ and the new structure observed by Belle.

Comments: 21 pages, 3 figures. Version to appear in PLB